Korean law states that all businesses over 10 people must pay Pension & Health Care. The law, as it is publicly available in English, is not ambiguous and doesn't leave much room for interpretation.

My Hagwon does not pay taxes. My hagwon does not participate in NPS. My hagwon does not file for Health Care. My Hagwon tells me that they are EXEMPT because the owner has 4 businesses, and there are loopholes in the law. ...the only thing I can can imagine is that if her 4 businesses have an average of under 10 workers per business, then she might be exempt, however I am not quick to believe that since we have had group trips before and it appeared that there might be more than 40 of us. There were, in fact, a busload of us.

What experience do we have here? I plan on going to the NPS this week to get an opinion from them, however even my girlfriend is telling me that it's likely futile and will cost too much money to recover the money.

Has anyone challenged their hagwon when they didn't pay the NPS, and has anyone ever had to deal with the terms they're giving me?

I believe you are looking at an old law. The pension applies to all workers now (the less than 10 worker exemption has been phased out I believe). Also, it never applied to foreigners anyways, because the Social Security agreement between Korea and most western countries makes it compulsory for a foreigner to have national pension.

The only way a Hagwon can skirt the law is by registering you as an independent contractor, instead of an employee. In this case, you must self-register and pay the full employer/employee premium yourself.

You can complain. The pension office will ask for details about your, salary, hours, relationship with the company you work for, to determine whether or not you are an employee under pension law and entitled to be enrolled by the workplace. Your workplace can be fined for failing to enroll employees into the pension scheme.